Many advanced control techniques are formulated as optimization problems, which can be solved by mathematical programming. One class of such techniques is optimization-based receding horizon control, such as model predictive control (MPC). There are MPC formulations for both linear and nonlinear systems. Nonlinear MPC solves nonlinear mathematical programs in real-time, which can be a challenging task due to a limitation on computing resources, the complexity of the problem to solve, or the time available to solve the problem. Therefore, most of the practical applications are based on a linearity assumption or approximation. The linear MPC typically solves a quadratic programming problem.
The MPC is based on an iterative, finite horizon optimization of a model of a machine and has the ability to anticipate future events to take appropriate control actions. This is achieved by optimizing the operation of the machine over a future finite time-horizon subject to constraints, and only implementing the control over the current timeslot. For example, the constraints can represent physical limitation of the machine, legitimate and safety limitations on the operation of the machine, and performance limitations on a trajectory. A control strategy for the machine is admissible when the motion generated by the machine for such a control strategy satisfies all the constraints. For example, at time t the current state of the machine is sampled and an admissible cost minimizing control strategy is determined for a relatively short time horizon in the future. Specifically, an online or on-the-fly calculation determines a cost-minimizing control strategy until time t+T. Only the first step of the control strategy is implemented, then the state is sampled again and the calculations are repeated starting from the now current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control.
The MPC can be used to generate the actual trajectory of the motion of the machine based on a model of the machine and the desired reference trajectory by solving an optimal control problem over a finite future time horizon subject to various physical and specification constraints of the machine. The MPC aims for minimizing performance indices of the machine motion, such as the error between the reference and the actual motion of the machine, the machine energy consumption, and the induced machine vibration.
However, due to the receding horizon and tracking nature of the problem, the MPC has no guarantee of finding an admissible solution. Specifically, the existence of the solution on a certain window of data (horizon) does not by itself guaranty that when the data window is shifted, a solution still exists. Thus, the control that is optimal for one iteration at one time period can place the machine in the state that forces the machine to violate some constraints during the next iteration, see, e.g., a method described in U.S. Pat. No. 7,376,472
To address this problem, the conventional solutions either determine the optimal trajectory off-line to allow the time to verify the trajectory, or determine a suboptimal, but feasible trajectory as an input to the optimization method, or increase a length of the finite future time horizon to reduce the uncertainty of the optimization, see, e.g., Mayne et al.: “Constrained model predictive control: Stability and optimality.” Automatica, Volume 36, Issue 6, Pages 789-814, June 2000.
However, in a number of applications, it is not possible to determine the feasible trajectory off-line. In addition, the increase of the finite future time horizon makes the computation more difficult, which can prevent the computation of the control strategy to be completed in the time allowed for the MPC method. Also, for longer time horizon, the future information required to determine the feasible trajectory might not be available.
Some conventional methods guarantee admissibility with respect to certain tracking performance metrics only at the steady state operation of the controlled machine. For example, Pannocchia: “Disturbance models for offset—free model—predictive control,” AIChE Journal (2003), 49(2), 426-437, Maeder et al.: “Linear offset-free model predictive control,” Automatica (2009), 45(10), 2214-2222, describe control methods that guarantee tracking a reference trajectory at a steady state. Also, Ferramosca et al.: “MPC for tracking of constrained nonlinear systems,” CDC/CCC 2009 and Falugi et al.: “Tracking performance of model predictive control,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on (pp. 2631-2636) describe methods that modify the reference trajectory to satisfy constraints on the machine dynamics.
All these methods fail to guarantee any performance metrics of the tracking of the trajectory during the transient phase of the operation. However, the transient response of a machine to a change from equilibrium is important for a number of applications, such as factory automation and automotive or aerospace vehicle controls. It is believed, that currently there is no method in the art that can guarantee the feasibility of tracking performance metrics of the machine tracking the trajectory in the transient stage.